The Convolution Method

Two dice. Probability the sum equals 7? Enumerate 36 outcomes. Three dice: 216. Ten dice: 60 billion. Convolution gives the exact distribution of the sum of independent random variables, no approximation. On FAM, problems are designed for hand computation.

Calculate probabilities by convolution when independent components are added. For each possible value of the sum, enumerate the component pairs that produce it and add their joint probabilities.

HIGH-FREQUENCY: Discrete convolution for two independent random variables is the most testable form on FAM.

If $X$ and $Y$ are independent discrete random variables:

$$\Pr(S = s) = \sum_k \Pr(X = k) \cdot \Pr(Y = s - k)$$

Sum over all $k$ where both terms are positive.

KEY: Convolution requires independence. You can only use $\Pr(S = s) = \sum_k \Pr(X = k) \cdot \Pr(Y = s - k)$ when $X$ and $Y$ are independent. Without independence, you need the joint distribution.

Common mistakes

• Forgetting contributions from multiple values of $N$. $S = 2$ arises from $N = 1$ and $N = 2$. Missing the $N = 2$ term gives 0.090 instead of 0.139.
• Omitting the $N = 0$ term. When $\Pr(N = 0) > 0$, there is a point mass at $S = 0$. Forgetting it makes probabilities sum to less than 1.
• Double-counting symmetric pairs. (1, 3) and (3, 1) are different events (count both. But (2, 2) is one event) do not count twice.

Bottom line

• Discrete convolution: $\Pr(S = s) = \sum_k \Pr(X = k) \cdot \Pr(Y = s-k)$
• Convolution yields exact distributions with no approximation, unlike normal or lognormal methods.
• For more variables: convolve iteratively, first $X_1 + X_2$, then add $X_3$.
• Collective risk model: condition on $N$, then compute the n-fold convolution for each $n$.

Exam shortcut

If the question asks for a single $\Pr(S = s)$, enumerate only the pairs that produce $s$. Do not compute the full distribution. If the question asks for $\Pr(S \leq s)$ or the complete distribution, build the table from smallest to largest and verify the total is 1. "Slide and multiply" (fix one row, slide the other, multiply aligned entries. "Condition, convolve, combine") the three C's for compound distributions.

The full lesson (about 1,363 words, 9 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

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